div (øu) = u · grad o + o div u
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Let \( \phi = \phi(x) \), \( \mathbf{u} = \mathbf{u}(x) \), and \( \mathbf{T} = \mathbf{T}(x) \) be differentiable scalar, vector, and tensor fields, where \( \mathbf{x} \) is the position vector. Show that
![The image contains the formula for the divergence of a product of a scalar field \( \phi \) and a vector field \( \mathbf{u} \):
\[ \text{div} (\phi \mathbf{u}) = \mathbf{u} \cdot \nabla \phi + \phi \, \text{div} \, \mathbf{u} \]
**Explanation:**
- **\(\text{div} (\phi \mathbf{u})\)**: Represents the divergence of the product of \( \phi \) (a scalar field) and \( \mathbf{u} \) (a vector field).
- **\(\mathbf{u} \cdot \nabla \phi\)**: Represents the dot product of the vector field \( \mathbf{u} \) and the gradient of the scalar field \( \phi \).
- **\(\phi \, \text{div} \, \mathbf{u}\)**: Represents the product of the scalar field \( \phi \) and the divergence of the vector field \( \mathbf{u} \).
This identity is useful in vector calculus when dealing with fluid dynamics, electromagnetism, and other fields involving differential equations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fedadeb2e-cdac-43d1-98ad-015a8ab931d0%2Fd1b26b39-fdc2-4fd4-a86c-4ba059e4af8e%2Fl7djye_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The image contains the formula for the divergence of a product of a scalar field \( \phi \) and a vector field \( \mathbf{u} \):
\[ \text{div} (\phi \mathbf{u}) = \mathbf{u} \cdot \nabla \phi + \phi \, \text{div} \, \mathbf{u} \]
**Explanation:**
- **\(\text{div} (\phi \mathbf{u})\)**: Represents the divergence of the product of \( \phi \) (a scalar field) and \( \mathbf{u} \) (a vector field).
- **\(\mathbf{u} \cdot \nabla \phi\)**: Represents the dot product of the vector field \( \mathbf{u} \) and the gradient of the scalar field \( \phi \).
- **\(\phi \, \text{div} \, \mathbf{u}\)**: Represents the product of the scalar field \( \phi \) and the divergence of the vector field \( \mathbf{u} \).
This identity is useful in vector calculus when dealing with fluid dynamics, electromagnetism, and other fields involving differential equations.
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